Erratum to: ‘‘Homotopy in functor categories”

Abstract

J*LanjJ*X J*LanJA = C o (JOP X J) cA ?J*X kJ*9 J*X J*X is a 2-cofibration. Since J* is a left adjoint, the lower square is a pushout. Since (J*eX)71J*X = 1, the conclusion follows in routine fashion. The additional hypothesis asks to be characterized as the assertion that C' C C is a cofibered subcategory. There have been other uses of the adjective in category theory, but they should lead to no confusion in this context. The rest of ?3, including especially Theorem 3.5, evidently requires this stronger hypothesis as well, so that the results below on homotopy relative to a subcategory are to be asserted only when the subcategory is cofibered. This restriction is real, but perhaps not unduly onerous. Its scope is indicated by the following two examples: (i) In general, C0 C C is cofibered if and only if, for each object c, 1C C(c, c) is cofibered in T5. (ii) If C is CY-discrete then C' C C is cofibered if and only if, for any composition xyz in C with x, z and xyz in C', y must also be in C'. Thus, for example, if C is the free group on one generator t and C' the submonoid generated by t then C' is not cofibered in C. Taking X = 0, A = {x, y}, tx = ty = x, we get a counterexample to the original (false) statement of Lemma 3.1.